AI 中文总结
本文研究度约束下Leaky-ReLU网络的最小节点可观测性与可控性，通过图论分析确定其最小观测节点数的最坏与最优精确值，利用对偶关系得到最小控制节点数的对应结果，并对比ReLU网络分析斜率的影响。
AI 中文摘要
本文研究了度约束下带泄漏修正线性单元（Leaky-ReLU）网络的最小节点可观测性与可控性，目标是确定需要测量或直接驱动多少个状态节点，才能从有限输出序列确定初始状态，或在有限时间范围内将网络在任意状态间引导。对于可观测性，图论分析给出了该类网络最小观测节点数的通用上界，本文构造了达到该上界的网络族，从而确定了最坏情况下最小观测节点数的精确值；还构造了可在有限时间范围内从单个节点观测的网络，确定了最优情况为1个节点。通过建立对应度约束下的可观测性-可控性对偶关系，本文得到了最小控制节点数的类似最优与最坏情况精确结果。与ReLU网络的比较显示，将负斜率为零替换为非零斜率会改变观测节点的需求；更一般地，可观测性论证仅要求激活函数的单射性，而可控性结果可扩展到双射激活函数。
英文摘要
This paper studies minimum-node observability and controllability of Leaky rectified linear unit (Leaky-ReLU) networks under degree constraints. The objective is to characterize how many state nodes must be measured or directly actuated to determine the initial state from a finite output sequence or to steer the network between arbitrary states within a finite horizon. For observability, a graph-theoretic analysis yields a class-wide upper bound on the minimum number of observation nodes. We construct a family of networks attaining this bound, thereby determining the exact worst-case minimum number of observation nodes. We also construct networks that are observable from a single node over a finite horizon, establishing the exact best-case value of one. By establishing an observability--controllability duality under the corresponding degree constraints, we obtain analogous exact best- and worst-case results for the minimum number of control nodes. A comparison with ReLU networks shows how replacing the zero negative slope with a nonzero slope changes the observation-node requirement. More generally, the observability arguments require only injectivity of the activation function, whereas the controllability results extend to bijective activation functions.
